Equation of Astroid/Cartesian Form

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Theorem

Let $H$ be the astroid generated by the rotor $C_1$ of radius $b$ rolling without slipping around the inside of a stator $C_2$ of radius $a = 4 b$.

Let $C_2$ be embedded in a cartesian coordinate plane with its center $O$ located at the origin.

Let $P$ be a point on the circumference of $C_1$.

Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis.


Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.


The point $P = \tuple {x, y}$ is described by the equation:

$x^{2/3} + y^{2/3} = a^{2/3}$


Proof

By definition, an astroid is a hypocycloid with $4$ cusps.


Astroid.png


From the parametric form of the equation of an astroid, $H$ can be expressed as:

$\begin{cases} x & = 4 b \cos^3 \theta = a \cos^3 \theta \\ y & = 4 b \sin^3 \theta = a \sin^3 \theta \end{cases}$


Squaring, taking cube roots and adding:

\(\displaystyle x^{2/3} + y^{2/3}\) \(=\) \(\displaystyle a^{2/3} \paren {\cos^2 \theta + \sin^2 \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle a^{2/3}\) Sum of Squares of Sine and Cosine

$\blacksquare$


Sources